Abstract
We construct a Generalized p value for testing statistical hypotheses on the comparison of mean vectors in the sequential observation of two continuous time multidimensional Gaussian processes. The mean vectors depend linearly on two multidimensional parameters and with different conditions about their covariance structures. The invariance of the generalized p value considered is proved under certain linear transformations. We report results of a simulation study showing power and errors probabilities for them. Finally, we apply our results to a real data set.
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Notes
For details, see Johnson et al. (1994).
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Acknowledgements
The authors thank the anonymous referees for their helpful comments and suggestions. The first and third authors were supported by MTM2015-65825-P
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Appendices
Appendix 1
Following an approach similar to that of Gamage Gamage (1997), if hypothesis testing is \(H_0:\delta =\delta _0\) versus \(H_1:\delta \ne \delta _0\), we will replace (in the previous shown material) \(T_{st}\) by \((T_{st}-\delta _0)\), and we obtain
where
If we consider the following hypotheses
we replace the \(T_{st}\) from earlier by \(l'(T_{st}-\delta _0)\), and we have as a result that
Then we have replace \(\sigma _t^{(1)}\) by \(l'\sigma _t^{(1)}l\), and \(\sigma _s^{(2)}\) by \(l'\sigma _s^{(2)}l\), and we obtain
We can rewrite \(W_{st}^{\delta _0,l}\) as following
with
Let us observe that the conditions \(\delta =\delta _0\) and \(l'\delta =l'\delta _0\) are equivalent, for every non null vector l.
Next, let
Considering that
and that this maximum is attained at \(x=cB^{-1}d\), \(\forall c\ne 0\), then \(t_l^2\) will be the maximum for \(l=c(\widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)})^{-1}\left( \widetilde{T}_{st}-\delta _0\right) \), that is, \(l=d_{st}^{\delta _0}\) maximizes \(t_l^2\), and the maximum is
so,
For \(l=d_{st}^{\delta _0}\), we obtain
As a conclusion, we obtain that the contrasts \(H_0:\delta =0\) and \(H_0':d_{st}'\delta =0\) or equivalently \(H_0': t_{st}^2(0)=0\), where \(t_{st}^2(\delta )=t_{st}^2(d_{st},\delta )\), are equivalent; indeed we accept \(H_0'\) if the observed value \(w_{st}^{0,d_{st}}=d_{st}'\widetilde{T}_{st}\) is small, that is, if \(w_{st}\le c^2\) then
This previous inequality implies that we accept the hypothesis \(l'\delta =0\), for all l, so we accept the hypothesis \(H_0:\delta =0\). On the other hand obviously to accept \(H_0:\delta =0\) implies to accept \(H_0': d_{st}'=0\). Similarly the following contrasts are equivalent:
-
(1)
\(H_0: \delta =\delta _0\) against \(H_1:\delta \ne \delta _0\).
-
(2)
\(H_0':d_{st}'\delta =d_{st}'\delta _0\) against \(H_1':d_{st}'\delta \ne d_{st}'\delta _0\) or equivalently \(H_0':t_{st}^2(\delta )=t_{st}^2(\delta _0)\).
Thus, we can conclude (for the initial contrast \(\delta _0=0\)) that
Appendix 2
Generalized p value for the unilateral case
Now, we consider \(H_0: \delta =0\) versus \(H_1:\delta >0\). As we can see in the Appendix 1, for \(W_{st}\) obtained for the bilateral hypothesis test, we use the direction \(d_{st}=(\widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)})^{-1}\left( \widetilde{T}_{st}-\delta _0\right) \) derived from \(\max _{x\ne 0}\frac{(x'd)^2}{x'Bx}\). Regarding the unilateral test, Park (2010) used the direction provided from \(\max _{x\ge 0}\frac{x'd}{\sqrt{x'Bx}}\) since it seems to be that \(d_{st}\) may be inflated the Type I error rate in the unilateral case, as we can see in a simulation study.
Following the approach presented in Park (2010), we have that
thus \(\widetilde{T}_{st}\) it will be a normal multivariate r.v. with mean \(\delta \) and variance-covariance matrix \(\widehat{\Sigma }=\widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)}\). So, we can write the following expressions
We define the generalized p value for the one-sided test as follows:
In order to evaluate the previous result, we realized some simulations with R software, as Park (2010, p. 1050) did in his study, using approximations presented in formula (27), (28) and (30). The results for power for one-sided test when \(k=3\) are shown in Table 5.
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Fenoy, M., Ibarrola, P. & Seoane-Sepúlveda, J.B. Generalized p value for multivariate Gaussian stochastic processes in continuous time. Stat Papers 60, 2013–2030 (2019). https://doi.org/10.1007/s00362-017-0907-7
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DOI: https://doi.org/10.1007/s00362-017-0907-7